How to intersect a texture with another mesh?
In order to replicate the effect , as shown by the question you have given above , in 2d with textures you will have to use a different approach, which is by using the blender's node editor,  the texture (here, logo) you want to apply the effect on should be added as an image texture in the node editor and then you will have to add a mix node and mix it with a transparent shader. Now, you can create a gradient and use a color ramp to tighten it up, also connect a Voronoi texture in between the texture node and mapping node in order to create a river like effect and use the x or y co-ordinates of the mapping node to make it go all the way from black to white ( if you want to inverse it just make it go from white to black and add an invert node after the color ramp ) and use it as a factor for the mix node. Finnaly you are going to achieve the effect you are looking for.